ASTR 5110 (Majewski) Lecture Notes

Observing Through Earth's Atmosphere:
Turbulence and Seeing

"If the Theory of making Telescopes could at length be fully brought into Practice,
yet there would be certain Bounds beyond which telescopes could not perform. For the Air
through which we look upon the Stars, is in perpetual Tremor; as may be seen by the
tremulous Motion of the Shadows cast from high Towers, and by the twinkling of the
fix'd Stars... The only Remedy is a most serene and quiet Air, such as may perhaps be
found on the tops of the highest Mountains above the grosser Clouds."
Isaac Newton

Section 1.3 of Bely, The Design and Construction of Large Optical Telescopes.

Section 5.8.1 of Hecht, Optics.

Chapter 5 by Quirrenbach in Lawson (ed.), Principles of Long Baseline
Stellar Interferometry.

Atmospheric turbulence is produced by a cascading transfer from larger to
smaller eddies, all which contribute to poor seeing
(van Gogh's Starry Night).

A. Introduction to the Problem of "Seeing"

All ground-based astronomical observations are hobbled by the fact that the light
must pass through the Earth's atmosphere.

Since the atmosphere is layered by (or consists of varying gradients in)
temperature and pressure, it has refractive power.
We have discussed the effects of the atmosphere's global refractive power elsewhere.

Worse than the presence of its net global refractive power is the
fact that atmospheric layering is not smooth.

Wind and convection and other currents
create turbulence, which mixes layers with
differing indices of refraction in non-uniform and constantly changing ways.

The net result has a serious effect (e.g., tilting, bending and corrugating)
on transiting, initially plane-parallel wavefronts.

From Schroeder, Astronomical Optics.

This is the source of the "twinkling" phenomenon we are familiar with.

The observed negative impacts of the turbulent atmosphere on astronomical images are encompassed
globally under the expression "seeing".
Understanding the physics of seeing allows us to:

Improve site selection of telescopes for better image quality/stability.

Improve the design of observatories to reduce the local effects of seeing.

Improve the conditions at existing observatories by reducing the local effects of seeing.

B. Physics of Turbulence and Seeing

Index of Refraction of the Air
We have previously adopted the index of refraction of air as n=1.
But in fact the index of refraction of air has a small variability, depending on
its physical state and composition.
The variable part of the refraction index is given by
Cauchy's formula (extended by Lorenz to account for humidity):

where λ is the wavelength of light, p is the atmospheric pressure
(mbars), T is the absolute temperature (K) and v is the water vapor pressure
(mbars).

The dominant terms in this equation translate to :

n - 1 = 77.6 X 10-6p / T

To give some sense of the small variability in this index,
for 500 nm light, n = 1.0003 at sea level and n = 1.0001
at 10 km altitude.

In terms of seeing, what we care about are changes in this quantity affecting
the transiting wavefront, or, more insidiously, differentially affecting
the wavefront on small scales (turbulence).

Fluctuations in water vapor have no significant effect on the refractive
index at optical wavelengths, except in extreme situations, such as in fog
or just above sea surface.
Water vapor can affect the radiative transfer properties of the air, and
therefore alter the convective properties of air columns.
The effect of water vapor (latter term in equation above)
is generally small for modern astronomical
observatories, which are typically built in very dry sites, and which already typically have
other weather-related observing problems when the humidity is high (e.g., condensation
on mirror surfaces, etc.).

If one takes derivatives of the above equation with respect to temperature
and pressure, one finds that variations in temperature are far more important
than variations in pressure, and, assuming adiabatic conditions and a perfect gas

dN / dT ∝ (p / T 2)

Thus, the primary source of variations in the index of refraction
are attributable to thermal variations.

In terms of seeing, what we care about is small scale variations in T,
or thermal turbulence.

Sources of Thermal Turbulence
Atmospheric turbulence is created on a variety of scales from several origins:

Convection: Air heated by conduction with the warm surface of the
Earth becomes buoyant and rises into cooler air, while the cooler air descends.

Wind shear: High winds, particularly the very fast ones associated with the jet stream,
generate wind shear and eddies at various scales, and create a turbulent interface between
other layers that are in laminar (i.e., non-turbulent) flow.

Disturbances: Large landform variations can create turbulence,
particularly on the lee side of mountains where the air flow becomes very
non-laminar.

From Quirrenbach's chapter in Lawson (ed.),
Principles of Long Baseline Stellar Interferometry.

Turbulence and the Eddy Cascade
The properties of fluid flows are characterized by the Reynold's number,
a dimensionless quantity relating inertial to viscous forces

Re = V L / ν = (Inertial forces) / (Viscous forces)

where V is the fluid velocity, L is a characteristic length scale,
and ν is the kinematic viscosity (m2/s) of the fluid.

Re high --- dominated by inertial forces and therefore
turbulent and full of vortices and eddies.

For air, viscosity is very small: ν = 1.5 X 10-5 m2/s.

Thus, for typical wind speeds and length scales of meters to kilometers,
Re > 106 and the air is moving turbulently.

One can think of turbulence as being made up of many eddies of different sizes.
The nature of thermal turbulence is created by a process of Eddy transfer:

Kinetic energy is deposited into
turbulence starting with the large scale air flow
processes of convection or wind shear.

The characteristic scale over which the energy is
deposited, L0
is called the external scale or outer scale.
L0 is generally larger than the aperture of
a telescope, but there is considerable debate over its typical
value.
Somewhere between 1 to 100 meters.

The large turbulent eddies created by the above processes
create wind shears on a smaller scale.

These still smaller eddies, in turn, spawn
still smaller eddies, and so on, in a cascade to smaller and smaller scales.

From http://www.lsw.uni-heidelberg.de/users/sbrinkma/thesis/node5.html.

This intermediate range of cascading
turbulent scales is called the inertial range, and is where:

inertial forces dominate and energy is neither created or destroyed
but simply transferred from larger to smaller scales.

all of the thermal fluctuations relevant to seeing occur.

The cascade continues until the shears are so large relative to the
eddy scale (Re ~ 1) that the small viscosity of air
takes over and the kinetic energy is "destroyed" (converted into heat).

Happens on scale, l0 , of a few millimeters.

Called the internal or inner scale.

The cascade stops.

The temperature fluctuations are smoothed out.

A Model for Turbulence: The Kolmogorov Spectrum
It is perhaps somewhat surprising that for the inertial range there is a
universal description for the turbulence spectrum (the strength of the
turbulences as a function of eddy size, usually expressed in terms of
wave number κ).

In 1941, Kolmogorov found that in the above process of an eddy cascade,
the energy spectrum had a characteristic shape:
The inertial range follows a spectrum of κ-5/3 (the
Kolmogorov spectrum or Kolmogorov Law).

From Lena, Observational Astrophysics.

When turbulence occurs in an atmospheric layer with a temperature
gradient (differing from the adiabatic one) it mixes air of different
temperatures at the same altitude and produces temperature fluctuations.
Hence, the above spectrum also describes the expected variation of temperature
in turbulent air.

Aside: The three-dimensional version of the spectrum follows κ-11/3.

The Structure Function
The theory of atmospheric seeing was developed from the 1960s to 1980s and
turns out to work remarkably well as a physical description of the process.
To understand turbulence in the context of seeing, we need to translate the
above thermal turbulence spectrum into a spatial context -- i.e., how the
atmosphere is going to transform wavefronts.

Generally and conveniently described in terms of the
structure function -- a statistical measure of the
fluctuation over a spatial span of r.

In one dimension, the structure function is defined as:

where T(x) and T(x+r) represent the temperatures for
two points separated by r.
The brackets denote a mean squared difference in temperature.

For a Kolmogorov spectrum:

where CT2 is the temperature structure constant
(or coefficient).
The size of CT2 gives the intensity of the spatial
thermal turbulence.

Ultimately, of course, we want the spatial variation in the index of
refraction, and this can be described by an index of refraction structure
constant, now derived by using the relation for n = n(P,T) above:

Connection to Fried Parameter
As we shall show below, there is a characteristic transverse linear scale over which we can
consider the atmospheric variations to flatten out, and in which plane parallel
wavefronts are transmitted.
This scale is known as the Fried Parameter, r0 ,
and is central to a description
of the effects of turbulence on images of stellar sources (it fully describes the
PSF).

As one might expect, the quality of an image depends on how much
turbulence there is along a line of sight.

The Fried parameter can actually be well predicted by integrating
Cn2 along a line of sight though the atmosphere:

where z is the altitude, and γ is the zenith distance (the angle
between the line of sight and the zenith).

An example of a Cn2 profile. Integrating
this profile through the airmass allows one to derive the Fried parameter.
One sees the largest Cn2 are near the ground,
although certain layers of the atmosphere, such as the altitudes of the
jet stream (about 104 meters) also contribute large turbulence.
From Bely, The Design and Construction of Large Telescopes.

A larger Fried parameter is better, hence the inverse dependence on
Cn2.

Note the direct dependence on cos(γ), which shows that
the Fried parameter gets smaller as cos(γ) gets smaller, which happens
as one looks through more
atmosphere with increasing zenith angle.

The Fried parameter, ro
is inversely proportional to the size of the PSF, θ,
transmitted by the atmosphere.
Thus, remarkably, the FWHM of the observed PSF (in arcsecs)
can be predicted if one knows
how Cn2 varies along the line of sight.
From the previous equation, and assuming now that the seeing is given
by the diffraction limit of one Fried cell (see below)
so that θ ~ λ/ro ,
one gets:

Again, note the dependence on zenith distance (in the sense that the PSF is smaller for
smaller zenith distance) and wavelength (in the sense that
the PSF is smaller for longer wavelengths).

Or, for λ = 500 nm and typical mountain top observatory conditions:

Time Dependence of the Turbulence
At a given point in the atmosphere, the temperature is a random function of time.
We can characterize the time frequency character of seeing phenomena considering the above physical
description.

A simple model (the Taylor Hypothesis)
is to consider the turbulence along a line of sight as "frozen" with given spatial
power spectrum and configuration, and assuming that a uniform wind translates the
column of air laterally with a velocity V.

(In effect this is the basis of the Fried description of seeing we adopt below.)

The physical basis of the Taylor hypothesis is simply that the timescales involved
in the development of turbulence are much longer than the time for a turbulent element
displaced by wind to cross the telescope aperture.
Rather than think of "frozen turbulence", probably more correct to think of a "frozen
phase screen" over short timescales.

The temporal cut-off frequency (the quickest timescale for observed changes in
image deformation) can then be given by order of magnitude as:

fc = V / l0

For l0 = 10 mm and V = 10 m/s,
we get fc = 103 Hz.

Cross-correlations of the wavefront from a star reaching a telescope pupil at intervals of 40 ms.
The displacement of the "phase screen" is obvious.
From Lena, Observational Astrophysics.

In reality, it is found that the turbulence at any particular site generally consists of
a superposition of this "frozen" turbulence (generally from higher layers) and a local turbulence
which has a significant vertical component, generally from convection near the
telescope from telescope, dome, other local features, etc.
Still, adopting the Taylor hypothesis allows us to gain some important insights using the
Fried description, as we did above and will again below.

C. The Turbulent Layers of the Atmosphere

Overall Vertical Structure of Atmosphere
To first order, the Earth's atmosphere is in radiative equilibrium, with
a balance of the solar flux and the energy re-radiated into space.
But there are important cyclic variations (diurnal, annual) superposed.
The mean large-scale structure of the atmosphere shows variations in pressure and temperature:

The pressure drops off exponentially as:

P(z) = P0 e-z/H

where the scaleheight, H ~ 8 km near the ground.

From Lena, Observational Astrophysics.

The temperature structure is more complicated:

In the troposphere, dT / dz < 0 in general (it gets colder as
you go away from the ground).

Above this the tropopause, where dT / dz ~ 0.

Then enter stratosphere, where dT / dz > 0 in general
(it gets warmer as you go higher).

Note also:

Height of tropopause depends on latitude, almost touching "ground
level" over Antarctica (altitude of South Pole already 3 km, and
the scaleheight of the atmosphere more compressed there).

At all latitudes, and especially near ground level,
there can be significant deviations from average temperature distribution.

Of importance for our understanding of turbulence
are inversion layers, where dT / dz changes sign.

Also of importance (particularly near the ground)
is the strength of a temperature gradient compared
to the adiabatic gradient:

( dT / dz )ad
= - (Tm / H) (Cp - Cv ) / Cp

where:

H is the scaleheight

Tm is a mean temperature over that distance

Cp , Cv are specific heats of the air.

If a temperature gradient larger than the above is created there is
convective instability and vertical currents are created from
buoyant forces.

Atmospheric Layers in the Context of Seeing
When discussing seeing, there are four basic atmospheric layers of relevance, with gradual transitions
of properties between them:

The surface or ground layer is the layer within a few to several tens of
meters (depending on the site) above the ground.

Here turbulence is generated by wind shear due to frictional and topographic
effects at the ground surface.

Strongly influenced by site geometry and "roughness" of ground from boulders, crags,
trees.

Best sites have a surface layer as small as 5 meters (e.g., Paranal) or 6-10 meters
(Mauna Kea).

Pays to have telescope, enclosure, primary mirror above this layer.

The planetary boundary layer extends to of order 1 km above sea level.

Here there are still significant frictional effects from the Earth's surface,
but also significant vertical motion due to diurnal heating/cooling cycles of the ground
and air in contact with it.

Convective effects are in play up to the inversion layer, which tops the planetary
boundary layer, at about 1 km.

Modern observatories are generally situated above this layer (most desirable,
to get above planetary layer turbulence).

From Bely, The Design and Construction of Large Telescopes.

From Lena, Observational Astrophysics.

The atmospheric boundary layer is the next major air mass up, and is separated
from the planetary boundary layer by the thermal inversion layer.

No longer dominated by convection, but may still be effected by the ground, both
thermally and mechanically, e.g., by large mountains or ranges that poke through the
planetary boundary layer.

From Bely, The Design and Construction of Large Telescopes.

Gravity waves in air, like water waves on the ocean,
are waves in layers of a fluid medium with the property of a decreasing
density with altitudes and
where gravity provides a restoring
force (opposite buoyancy)
when a parcel of the fluid (air) is displaced, resulting in an oscillation
of the parcel about an equilibrium state.
An interesting video showing passage of gravity waves is shown
here.
Gravity waves transfer momentum from the troposphere to higher layers of the atmosphere,
with the amplitudes of the waves being larger in the more rarefied layers.

To avoid these long wavelength waves, best to be near the sea (over which
there is laminar flow) on side of prevailing winds (e.g., foothills of coastal ranges).

Best sites are also above the atmospheric boundary layer, but at many existing
sites, this is the dominant source of seeing.

E.g., at ESO site at La Silla, Chile, 80% of seeing is from the 500 m
of the atmospheric boundary layer still above the telescopes.

The atmospheric boundary layer above water generally exhibits smaller thermal
fluctuations than over land because the temperature contrast between water and air is
generally smaller.

Thus, observatory sites on islands or shorelines tend to be better.

Because lower layers of an airstream quickly adapt to land conditions,
close proximity to sea is best.

From Bely, The Design and Construction of Large Telescopes.

Were the jet stream the only consideration, then one would expect better seeing
(from the free atmosphere) near the equator and poles (but see future web page
on site selection for other considerations).
The contribution to seeing from free atmosphere is about 0.4 arcsec on average.

Diurnal Effects in the Low Altitude Layers
There are significant daily changes in the thermal profile of the lower atmospheric layers over
land, which
drive the changing convective properties within these layers.
The nature of this profile is important to the design of observatories, both night time and solar.
A typical example of the profile variation is as follows:

From Coulman, 1985, ARAA, 23, 19.

Solar heating of the Earth's surface during daytime leads to conductive heating of air in
contact with it.

By mid-afternoon, the heated air creates convective currents that extend the planetary boundary
layer to a typical maximum of 1.3-1.5 km, as shown by the purple, 3 PM curve above.

In the tropics or in disturbed weather, cumulus cloud growth leads to a
planetary boundary layer as high as 3 km or more.

Moist air is especially sensitive to vertical displacement, which can
induce condensation, clouds, precipitation.

The above represents an average profile, whereas large thermal fluctuations from this
average can occur on short timescales.

The thermal turbulence is much larger in convective updrafts than in downdrafts,
and so:

on average, the fluctuation amplitudes usually decrease with altitude.

but because there are typically bulk
updraft and downdraft cells, significant and quick
changes can occur in the seeing with the lateral shifting of the cells.

The Cn2 profile is generally given by the temperature variance
through the convective layers, and so, in principle, can be estimated.
A schematic view of thermal variance, incorporating mainly convective
effects, gives rise to something like this:

From Coulman, 1985, ARAA, 23, 19.

As seen in the above figure for daytime turbulence, it pays to locate a solar telescope as high
as possible above surrounding terrain, to get above the lower turbulence.

In fact, the actual Cn2 profile is more complicated than this.

Observations show the existence of thin layers having high wind shear or other
properties creating greatly enhanced Cn2.

From Coulman, 1985, ARAA, 23, 19.

The existence of these discrete, high Cn2 layers suggests
that seeing compensation (e.g., with adaptive optics) might be best accomplished by
optically complementing these discrete layers.

This is the basis of multi-conjugate adaptive optics (MCAO)
(to be discussed later).

The diurnal variation in the Cn2 profile near the
ground is shown by the thick solid lines for midday (line 3), late afternoon (line 2)
and a good seeing night (line 1).

A simplified version of Coulman's figure is shown in Bely:

From Bely, The Design and Construction of Large Telescopes.

Here is a version of this plot (with a linear altitude scale) -- showing primarily free atmosphere
effects above the VLT site at Paranal (which is above most of the other layers):

From Quirrenbach's chapter in Lawson (ed.),
Principles of Long Baseline Stellar Interferometry.

As pointed out in the caption, this particular data set does not show the typical
strong turbulence associated with the jet stream turbulence.

D. The Fried Model and Isoplanaticity

In 1665, Robert Hooke, in his Micrographia, first suggested the existence of "small,
moving regions of the atmosphere having different refracting powers which act like lenses" to
explain scintillation.
In 1966 David L. Fried showed that the atmosphere can indeed be modeled in this fairly simple
way.

One can assume that at any moment the atmosphere behaves like a compressed,
horizontal array of small, contiguous, wedge-shaped refracting cells.

These act on the plane parallel incoming waves from an astronomical source by
locally tilting the wavefront randomly over the size scales of the cells.
Each cell imposes its own tilt to the plane waves, creating local isoplanatic
patches within which the wavefront is fairly smooth (~ λ / 17)
and has little curvature.
Thus, each isoplanatic patch transmits a quality (though perhaps shifted)
image of the source.

From Kitchin, Astrophysical Techniques.

The critical scale over which images begin to lose quality (i.e., the
size of the isoplanatic patch) is called the Fried parameter or
Fried length and
is usually designated as ro .
This is not meant to be a radius, but is more like a diameter of the patch.
It describes the scale over which the wavefront maintains its parallel (though possibly
tilted) nature and the diameter over which bundles of rays still arrive parallel and
in phase.

The Fried parameter is a statistical description that characterizes the seeing.

The more turbulence there is, the smaller is ro and the poorer is
the seeing.

A typical value for the Fried parameter in the optical is about 10 cm.

However, the Fried parameter varies with wavelength as λ6/5.
This means that the seeing will always be better (i.e., PSF will always be sharper) at
longer wavelengths.

If we carry the model further, and assume that the turbulence is occurring in
a single layer, can view the changing aspects of seeing in terms of the speed that
wind moves the cells across the line of sight.
The coherence time -- the time for the transit of one cell in the Fried description
-- is then given by

τo ~ ro / v

From Bely, The Design and Construction of Large Telescopes.

The coherence time is typically 10's of Hz. (Note that this is longer than the "quickest" timescale
derived using the inner scale of the turbulence.)

Another important characteristic of seeing is the angular size of
the isoplanatic patch.
This isoplanatic angle is the
angle on the sky over which the effects of turbulence are uniform/correlated,
given by:

θo ~ 0.6 ro / h

where h is the altitude of the primary turbulence layer over the telescope.

E. Observed Seeing Effects: Scintillation, Image Wander, Image Blur

There are various manifestations of the observed effects of turbulence (seeing).
Scintillation changes the apparent brightness of a source, whereas image wander and image blur
degrade the long term image of a source.
Which of these effects comes into play depends on the telescope aperture size
relative to the Fried parameter.
ScintillationScintillation ("twinkling") is the result of a varying amount of energy being received by a
pupil over time.

Variations in the "shape" of the turbulent layer results in moments where it
mimics a net concave lens that defocuses the light and other moments where it is like
a net convex lens that focuses the light.
This curvature of wavefront results in moments of more of less light being received
by a fixed pupil.

Scintillation only is obvious when the aperture/pupil diameter is of order
ro or less.
E.g., the human eye pupil is generally always less than ro .
Larger apertures average out the affects and the "twinkling" is diminished.

Since scintillation is ultimately an interference phenomenon, it is highly
chromatic.
You may have noticed this by observing stars close to the horizon changing
color and strongly twinkling on timescales of seconds.

From Lena, Observational Astrophysics.

Image Wander
Another obvious effect for apertures smaller than ro is image wander
or agitation.

This is the motion of the instantaneous image in the focal plane due to changes
in the average tilt of the wavefront with time.

But because the wavefront is locally plane parallel, the image can actually
be diffraction-limited for each isoplanatic patch that passes through the line of sight.
Thus, a sharp Airy disk -- with central FWHM ~ λ / D --
can be formed from each ro cell,
and if the telescope aperture is smaller than ro,
it will be common to see a sharp Airy disk in the focal plane...
... but the center of that diffraction-limited Airy disk will wander around the image plane as
different cells imposing different wavefront tilts pass over the aperture.

From Roy and Clarke's Astronomy: Principles and Practice.

Thus, to instantaneously achieve the ultimate, diffraction-limited performance
of a telescope, one only needs to ensure that the telescope aperture be smaller than
ro (by about a factor of 1.6). Of course, this means you are forcing
yourself to lower light-gathering power and resolution!
This image wander is a characteristically observed phenomenon when looking through
small telescopes.
To obtain the diffraction limit over a long timescale,
one needs a guiding system that can remove the shifting image position...
... or a tip-tilt mirror system to correct out the wavefront tilts
(the basis of active optics seeing compensation -- see future lecture).
In this (relatively simple)
way the primary atmospheric degradation for small apertures can be removed.

From Bely, The Design and Construction of Large Telescopes.

Image wander diminishes as telescope aperture grows, because larger apertures
average over more isoplanatic patches.

Image Blurring
Things are more complicated for a large telescope (D > ro),
since many isoplanatic patches will be in the beam of the telescope, and
image blurring or image smearing dominates.

Because many parts of the corrugated wavefront are contributing, larger
telescope apertures suffer from a larger image spread from the accumulated
contributions of differently pointed, wavefronts.

At any given time, if looking at the image of a single star
in a large telescope, each isoplanatic patch creates its own
diffraction-limited Airy disk (FWHM ~ λ / D).
These individual Airy spots are called speckles.
Seen together, the speckles give a shimmering blur.

In this case, the ensemble of speckles will have an envelope given by
FWHM ~ λ / ro .

From Lena, Observational Astrophysics. Note the error in the figure
in that the number of areas of coherence is of course
N ~ (area of pupil) / (area of Fried cell),
or the inverse of what is shown.

If integrating over long times compared to the coherence timescale, the
resultant image, which is the superposition of all of the speckle patterns during the
integration, will have this size (FWHM ~ λ / ro)
for the long exposure seeing (PSF)
-- a significant degradation of image quality.

If the telescope diameter is of order or slightly larger than ro ,
you will see several speckles, but occasionally can chance upon a moment when the seeing
turns very good (ro is temporarily larger).

From Hecht, Optics.

In principle, one could fast shutter the telescope, leaving only these moments
of good seeing get to the detector and shuttering out the bad moments to get crisp
images.
But, as mentioned in the caption above, the larger the aperture, the likelihood
of a clear moment decreases exponentially with aperture diameter.

For a given small telescope of a set aperture size, what stellar images will look
like depends entirely on the seeing cell size compared to the telescope aperture.
For example, the figure below shows drawings of a star observed with the same small telescope
under different seeing conditions. As may be seen, under good conditions with ro
larger than the aperture we see an Airy pattern, as in the rightmost drawing, but as
ro shrinks to about the size of the telescope aperture, one sees something
more like the middle panel. Under terrible conditions, when ro is
much smaller than the telescope aperture, one can see speckles even in a small telescope
(with each speckle coming from a different Fried cell parcel of air).

From the Celestron CPC Series Instruction Manual.

In general, increasing the telescope aperture will collect more light (i.e. more speckles),
but will not improve the resolution (PSF) beyond the atmospheric, long exposure
seeing limit.

A strategy for taking advantage of the speckles to recover the theoretical
diffraction limit of a large aperture telescope is called speckle interferometry
(see next lecture webpage).

Note that because the Fried parameter itself goes as λ6/5, the
seeing PSF goes as λ-1/5 -- i.e., the seeing is always better in the
red.

Short and Long Exposure Seeing
Imagine that you could capture and pick apart the speckled image (and we will!) from one instant
in time (an exposure time a fraction of the coherence time).

Each speckle presents a diffraction-limited image of the target source.

Thus the MTF of each speckle contains high frequency
information to the diffraction limit ~ λ / D (solid line in the image below).
The solid line below then represents the short exposure seeing for a speckle
taken through a telescope of diameter D.

From Lena, Observational Astrophysics.

Imagine then that you collect an image from the telescope for much longer than the
coherence timescale.

This is like averaging over many of the short exposure speckle patterns.

We have seen above that the ensemble of speckles will have an envelope given by
FWHM ~ λ / ro .
Thus, the net PSF of the image of a star will have an extent given by this envelope,
FWHM ~ λ / ro .

As seen in the MTF above, the long exposure seeing profile (dashed line) will now have a cutoff
frequency given by ro / λ.
As the exposure time is lengthened past a fraction of the coherence timescale, higher
spatial frequency information begins to be lost (the region of the MTF between the dashed and
solid lines).

The corresponding comparison for what you see in the focal plane is shown below, with
a diffraction-limited image (as one would get from short exposure seeing) compared to
an image from long exposure seeing ("seeing disk").

If integrating over long times compared to the coherence timescale, the
resultant image, which is the superposition of all of the speckle patterns during the
integration, will have this size for the long exposure seeing (PSF)
-- a significant degradation of image quality.
From Roy and Clarke's Astronomy: Principles and Practice.

The image presented by the telescope with D > ro is that presented
by a telescope of diameter ro .
It is seeing-limited.

The size of the resulting, degraded image is called the seeing disk.
In this case, increasing the diameter of the telescope will not increase the resolution (decrease the size of the PSF)!

As we saw in the discussion of atmospheric layers above, much of the seeing is contributed by the
surface layer.

The improvement in the seeing as one moves away from the Earth surface is shown by the
long and short exposure MTFs presented below at 2, 10, 30 and 100 meters above the Earth surface.

From Coulman, 1985, ARAA, 23, 19.

The improvement in the cutoff frequency in the long exposure seeing is obvious.
There is no increase in the cutoff frequency for the short term seeing (it is the diffraction limit).
The tendency for the long term MTF/long term seeing to become more like the short term seeing is
one rationale for moving telescopes off the ground.

Note that the above figure is for daytime conditions when boundary layer convective
currents are worse, so the improvement is a bit overstated in the situation for night time
conditions.

Net Effects of Scintillation, Image Wander and Image Blur
Below is an enlarged image of the bright star Betelgeuse seen though a large
telescope.
Click on the image for a video of the seeing effects.
Note the numerous speckles, the image wander and imagine the
net image blur one would see in the long exposure image of this star.

Click on the above image to see "seeing" produced by Earth's atmosphere in
a series of quick snapshots. Note that in each snapshot you can see individual
speckles images of the star at the resolution limit of the telescope, similar
to the cartoon representation above.

A movie borrowed from the website http://lcogt.net/en/book/export/html/3684
showing the effects of bad seeing on images of the lunar crater Clavius.

Natural Site Seeing
When one discusses "the seeing" one is generally referring to the long exposure FWHM of the PSF.
One can characterize the typical image quality results by compiling statistics on the seeing
from night to night.

From Bely, The Design and Construction of Large Telescopes.

Seeing quality from Gemini South (Chile, left) and Gemini North (Mauna Kea, right).
From http://www.gemini.edu/metrics/seeing.html.

The seeing statistics can be converted to other useful image quality criteria:>

From Bely, The Design and Construction of Large Telescopes.

Many sites rely on special instruments called Differential Image Motion Monitors (DIMMs)
to measure their site seeing.

DIMMs traditionally measure the differential image motion (which are caused by wavefront
slope differences) between stellar images formed
between several sub-apertures of a small telescope, and use this to calculate the Fried parameter.

Small prisms help to separate the sub-pupil images in the focal plane.

Because this is a differential measurement, it is immune to wind shake, earthquakes (!)
and errors in tracking.

(Left) A view of the four sub-pupils for
a DIMM monitor made by the RoboDIMM company.
(Middle) The DIMMs are normally placed above the atmospheric ground layer in a small tower, such as this one
by the RoboDIMM company. (Right) A DIMM in place at La Palma (the building on the right, shown on
a peak overlooking another telescope).

A sequence of DIMM images of Aldebaran taken in 1.75 arcsec (poor) seeing
with a DIMM at the Telescopio Nazionale Galileo on La Palma.
The images, and how they are used to reconstruct the seeing, are taken from and described
here. The website also shows an image
of the dual pupil DIMM used to make the images.